Control Engineering

Feed forward or feed back?
May 29, 2007

The answer relates to an ancient riddle: “How many were going to St. Ives?” The full rhyme, attributable to that venerable and highly prolific authoress, Mother Goose, is:

As I was going to St Ives,
I met a man with seven wives
And every wife had seven sacks
And every sack had seven cats
And every cat had seven kits
Kits, cats, sacks, wives
How many were going to St Ives?

Wolfram MathWorld provides a rigorous mathematical treatment of the interesting part of the riddle (how many things I met), and points out that the rhyme’s origins probably go back to 1650 BCE, making Mother Goose a plagiarist. The website Rhymes.org.uk also provides some interesting background and characterizes the riddle as a means to teach children the art of “lateral thinking.”

Most intelligent people, when they first encounter this riddle, try to solve it by diving into the calculation a la MathWorld, and finding that summing the man I met, the wives, the sacks, the cats, and the kits totals 2,801 things. Usually, however, the questioner interrupts this fairly involved calculation by pointing out that the only one explicitly going to St. Ives is “I” (the questioner), so the answer is 1. Everyone then gets a good laugh at the expense of the poor sot who’s been fighting through the math.

I always found that answer way too glib. It, after all, makes the dazzling assumption that just because “I” was going to St. Ives, everyone I meet on the way must be NOT going to St. Ives. There is, in reality, no reason to believe that we couldn’t have meet on the road or at a rest stop, despite traveling in the same direction. It happens all the time. In fact, there’s nothing in the riddle that says we didn’t meet on purpose! It just says we met.

Considering that, the correct answer might be one, or 2,802.

But, wait! The rhyme says that the man had 7 wives and all the rest of that stuff, but it never says that he has any of them with him. He might have been traveling from (or to) St. Ives just to get away from all the ruckus 2,752 living creatures (2,801 minus the sacks) can raise at home. We might have had a rendezvous at a bar to commiserate over a beer. So, the answer may be 2 or 1.

Or, he could be waiting by the side of the road to meet the 2,800 wives, sacks, cats, and kits, and they were all going to travel to St. Ives in one, big, noisy group. How do you count that one? I guess that makes it 2,802 again.

Next, we really should look closely at the rhyme’s last two lines, which contain the question. Cast as single a prose sentence it reads: “Kits, cats, sacks, wives; how many were going to St. Ives?”

Note that the man and I are not on the list. Therefore, we don’t count. The answer could be 2,800 or zero.

Traditionally, people have assumed that the list in the penultimate line is unimportant window dressing, but if that were true, then there is nothing to qualify who counts and who doesn’t! Anyone traveling in the direction of St. Ives on that day might count. The answer could, therefore, take on any integer value between 1 and the entire population of medieval England—or the entire population of England minus 2,752.

In the end, one could reasonably interpret the riddle so that the correct answer turns out to be 0; 1; 2; 2,800; 2,801; 2,802; or any integer equal to or less than the population of England. There just isn’t enough information to clarify the situation.

In the end, the only correct answer is: “I don’t know.”

For engineers, that is the riddle’s significance. In many, if not most, situations, you just don’t know. You don’t have enough information. In fact, Paul Ormerod, in his book Why Most Things Fail, demonstrates quite well that in an amazingly high percentage of cases, not only do you not have enough information, but sufficient information simply does not exist to predict how a given system will behave in the future.

That’s why feedback control is almost always superior to feed forward control. Feed forward control makes it possible to react faster then feedback, but only if you can predict how your control inputs will modify the way the system was going to behave in the first place. If you can’t predict its behavior, then how can you predict what its behavior will be after your control inputs have modified its unpredictable behavior?

Feedback, on the other hand, makes no assumptions about the future. It makes a judgment call based on how things are right now. It also cancels the problem that system behavior is a moving target by the fact that “now” is a moving target, too.

Unlike feed-forward control, which assumes that the system will respond as it did in the past (when you worked out the control sensitivity matrix), feedback control allows for the possibility that things might change. It also automatically accommodates the “slings and arrows of outrageous fortune” that drove Hamlet near to suicide and can give control systems kinky fits.

Related Control Engineering articles of interest:
Auto-Tuning Control Using Ziegler-Nichols
Model Predictive Controller

Posted by on May 29, 2007 | Comments (0)